A symmetry of a quantum mechanical system is an invertible transformation of the state space that doesn't lead to observable differences in the statistics.
By Wigner's theorem [1]}, such symmetries are implemented by unitary (or anti-unitary) operators.
Due to this, continuous quantum mechanical symmetries are described by projective unitary representations of connected Lie groups.
These are unitary operators \(U_g\) , whose multiplication recovers the group operation up to a phase.
If the phase is trivial, one simply speaks of a unitary representation or of a proper unitary representation to emphasize.
In practice, projective unitary representations often occur if one exponentiates a representation of a Lie algebra:
A Lie algebra representation might not exponentiate to the group of interest but rather a covering group [2]}.
Take for example the spin-\(\frac{1}{2}\)  representation of \(\mathfrak {so}(3)\) .
Conceptually, it often makes more sense to view such a representation as a projective representation, i.e. a “representation up to phase”, of the group at hand, instead of viewing it as a a (proper) unitary representation of its cover.
In any case, the issue of additional phases only arises on the Hilbert space level.
For the induced unitary quantum channels \(\mathcal {U}_g=U_g(\,\cdot \,)U_g^*\) , the phases from \(U_g\)  and \(U_g^*\)  cancel each other.
